The Introduction of Irrational Numbers

Unsymmetrical Approximation of Irrational Numbers

1 A 1 < £ < , 5B B SB so that we have the classical theorem of Hurwitz. For other values of r, approximations from both sides are permitted, but the errors allowed on the two sides are different; hence the term unsymmetrical approximation. The result here was new, and is so related to Hurwitz's inequality that one side is strengthened and the other weakened. Notice that the result for r > l is ...

Weierstrass’s Construction of the Irrational Numbers

We present an overview of the development of the irrational numbers due to Karl Weierstrass. This construction was first presented during lectures in the 1860s in Berlin. Weierstrass never published his construction. Several of his students (Kossak, Horwitz, von Dantscher and Pincherle, to name a few) gave accounts in lecture notes from the courses. However, these notes were not published under...

On Irrational and Transcendental Numbers

Representing numbers: prime and irrational

This article draws an analogy between prime and irrational numbers with respect to how these numbers are defined and how they are perceived by learners. Excerpts are presented from two research studies: a study on understanding prime numbers by pre-service elementary school teachers and a study on understanding irrational numbers by pre-service secondary school teachers. Considering the results...

Continued Fractions, Fibonacci Numbers, and Some Classes of Irrational Numbers

In this paper we define an equivalence relation on the set of positive irrational numbers less than 1. The relation is defined by means of continued fractions. Equivalence classes under this relation are determined by the places of some elements equal to 1 (called essential 1’s) in the continued fraction expansion of numbers. Analysis of suprema of all equivalence classes leads to a solution wh...